On a infinite-dimensional group over a finite field
Anatoly M Vershik1998, Functional Analysis and Its Applications
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Abstract
Introduction. The asymptotic theory of representations studies the behavior of representations of classical groups of high degrees and their infinite-dimensional analogs. One of the key examples is the series of linear groups GL,~(Fq) over a finite field k = Fq, and the inductive limit GLc~(Fq) of these groups, the group of infinite invertible matrices over the field k that differ from the identity matrix by only finitely many nonzero entries. In this note we introduce a new locally compact group GLB and study its structure, characters, and continuous unitary representations. The group contains GL~(k) as a countable dense subgroup. The importance of the group GLB relies on its close connections with the theory of parabolic induction, which is the core of the representation theory of the groups GL,(k). The description of irreducible representations of the groups GL,(k) was first obtained by Green [6]. The main tool of his theory, which was developed later in the papers [5, 10], is the operation of parabolic induction of representations. Just this operation determines the inclusions of group algebras C(GL,,(k)) that lead to the group algebra A of the group GLB to be studied in this paper. The algebra A can be represented as an inductive limit of the algebras C(GLn(k)), though the inclusions of these group algebras are determined by averaging operators over the cosets of a subgroup, rather than generated by group inclusions. The group GLB has been defined by the authors of the present paper together with A. V. Zelevinsky (see the supplement by the editor of the Russian translation of [3]) in 1981 in the course of discussing the relationships of his paper [10] with the theory of representations and characters of locally finite groups (similar to GLoo(k) and to the infinite symmetric group 6~). Our approach relies on the general theory of inductive limits of finlte-dimensional semi.qimple algebras. Note that, for q-1, we obtain the representation theory of the infinite symmetric group; this theory was studied in a n11rnber of papers by the authors (see, e.g., the references in [1]). The remarkable fact that our infinlte-dimensional nondiscrete group GLB is locally compact allows one to use the full power of classical methods of representation theory and, in particular, the method of induced representations. We intend to study the unipotent characters of the group GLB, and give realizations of factor representations of the groups GLB and GLoo (k) in detail in forthcoming papers. 1. The group GLB. The objects defined below depend on the cardinality q of the field but, for convenience of the notation, we do not always show this dependence explicitly. Consider the k-linear space V-k~ of all finite vectors with coordinates in a finite field k = Fq. Fix a basis of vectors {ei, i = 1, 2,... } in the space V and denote by Vn the subspace generated by the first n vectors. Definition. The group GLB consists of k-linear transformations of the space V that preserve all subspaces Vn except for finitely many of them. Let us reformulate the definition in terms of matrices of operators g E GLB with respect to the distinguished basis. An infinite matrix g = (g~j), i, j = 1,2,..., is said to be almost triangular if the number of its nonzero subdiagonal elements a~i r 0, i > j, is finite. The group GLB consists precisely of operators with almost triangular matrices.
